# Gennaro Miele » 3.Expansion stages

### The flatness problem

The hot Big Bang model suffers from some problems which are related to the kind of expansion, and particularly if one assumes that the history of the Universe has been always characterized by a decelerated expansion. Some of these problems depend on the theoretical framework chosen, whereas others require extremely well tuned initial conditions which sound very unnatural.

The flatness problem is related to the curvature contribution to the Universe expansion, parameterized by Ωk, which increases quickly with time if

$\ddot{a}<0$

From the Friedmann equation we have

$\left| \Omega -1 \right| = \frac{|k|}{a^2 H^2} \sim |k| a^{1+3 w}$

where ω is the total pressure-to-density ratio. For a decelerated expansion (like for instance during radiation and matter domination) one has 1+3  ω>0. In this case Ω = 1 is a past actractor, so going back in time FRW metric is closer and closer to the flat limit. Nevertheless, today we measure a density extremely close to the critical value, and this means a fine-tuned curvature radius in the early Universe down to unnaturally small values.

### The horizon problem and the inflation stage

The horizon problem is due to the fact that if the expansion was always decelerated in the early universe (like during matter or radiation dominated phases), the particle horizon at the time of decoupling zLS would be too small. This scale should be seen on the last scattering surface under an angle of approximately 10. Larger angular scales should correspond to patches which were causally disconnected at photon decoupling, so it is paradoxical that the temperature is the same on the whole solid angle, up to tiny fluctuations of the order of  ΔT ≈ 10-5 T0. Let us briefly summarize the point. The farthest distance, measured at a time t, that a light signal could have traveled since the Big Bang (conventionally placed at t = 0) is the so called physical horizon

$d_H(t) = a(t) \int_0^{a(t)} \frac{d a' }{a'^2 H(a')}$

that for a matter dominated Universe ( namely a(t) ≈ t2/3 ) becomes dH(t) = 3 c t . Note that the matter domination epoch represents the main part of thermal history of the Universe. The CMBR that we measure today is coming uniformly from every direction of the sky. The physical distance of the source of CMBR at the time of its emission was

$d_{CMB}(t_e) = a(t_e) \int_{a(t_e)}^{a(t_0)} \frac{d a' }{a'^2 H(a')}$

where a(te) and a(t0) are the Universe scale factor at CMBR emission and today, respectively.

### Causally connected regions at the time of CMB

At the time of emission, the sources of radiation coming from opposite directions of the sky would have been separated by a physical distance of about ds = 2 dCMB . To understand if regions emitting CMBR would have been in causal contact with each other one has to compare dwith the horizon size at that time. If this ratio is >> 1, the two regions are outside each other’s horizon$d_s(t_e) = 2 a(t_e) \int_{a(t_e)}^{a(t_0)} \frac{da'}{a'^2 H(a')} = \frac{6 c t_0 a(t_e)}{a(t_0)} \left(1 - \left( \frac{t_e}{t_0}\right)^{1/3} \right)$              $d_H(t_e) = a(t_e) \int_{0}^{a(t_e)} \frac{da'}{a'^2 H(a')} = \frac{3 c t_0 a(t_e)}{a(t_0)} \left( \frac{t_e}{t_0}\right)^{1/3}$

and hence the ratio results

$\frac{d_s(t_e)}{d_H(t_e)} = 2 \left(\left(\frac{t_0}{t_e} \right)^{1/3} - 1 \right)$

observing that in terms of redshift ze at the emission, namely 1 + ze =a(t0)/a(te) = (t0/te)2/3,

$\frac{d_s(t_e)}{d_H(t_e)} = 2 \left(\left(1 + z_e \right)^{1/2} - 1 \right)$

since the emission was at ze ≈ 1500, this gives a separation ratio ≈ 80, hence absolutely incompatible with the high level of isotropy of CMBR.

### Inflation as a solution for flatness and horizon problems

Starobinsky and Guth proposed to solve flatness and horizon problems by assuming a stage of accelerated expansion prior to radiation domination epoch. This means a phase in which ω < -1/3.

Let us denote with ain and aend the values of the scale factor corresponding to the beggining and end of the accelerated phase, respectively. Let us also assume that the expansion is a quasi-de Sitter one, namely with ω ≈ −1, like it occurs in a cosmological constant dominated Universe. This stage would contribute to the particle horizon by an amount

$d_p = {a_{end}} \int_{a_{in}}^{a_{end}} \frac{d a' }{a'^2 \overline{H}} = \frac{a_{end}}{\overline{H} a_{in}}-\frac{a_{end}}{\overline{H} a_{end}} \sim \frac{a_{end}}{\overline{H} a_{in}}$       with $\overline{H}$  the (almost) constant value of the Hubble parameter during inflation. The last equality holds if inflation lasts long enough, so that aend >> ain. In this case we see that the particle horizon grows like the scale factor. If we start with a small initial patch at ain, inside which causality is established by physical processes, this single patch blows up and might embed the whole present observable Universe, or even a much bigger region. Then all observed CMB photons may originate from a single causally connected region, and there is no more horizon problem. The flatness problem also disappear since accelerated expansion with ω <-1/3 implies that Ω is driven extremely close to one during inflation. Another crucial property of inflation is that physical scales grow faster than the Hubble radius during any accelerated expansion stage, while during radiation or matter domination they tend to re-enter within the Hubble scale. Then the largest cosmological scales that we can observe today, that are of the order of present Hubble scale, might have been within a single causally connected region already at the beginning of inflation.

### Radiation dominated (RD) epoch

At the end of inflation, the inflaton decay reheated the Universe. At this time the energy density was dominated by the radiation density contribution ρR≈ k T4. Big Bang Nucleosynthesis provides a striking evidence of such period down to at least T ≈ MeV via the expression of the Hubble parameter that in RD epoch is proportional to (ρR)1/2 ≈ T2/MPl .  The energy density and pressure of the total radiation (ultrarelativistic particles) can be expressed in terms of the photon energy density for each polarization d.o.f. as

$\rho_R = g_* \frac{\pi^2}{30} T^4 = 3 P_R$

with

$g_* = \sum_{i, boson} \, g_i \left( \frac{T_i}{T} \right)^4+ \frac{7}{8}\sum_{j, fermion } \, g_j \left( \frac{T_i}{T} \right)^4$

where we have considered the possibility that the different particle species “i” do not share the same temperature. T stands for the photon temperature. Note that non relativistic d.o.f. as the temperature decreases above their mass value disappears with the Boltzmann factor e-m/T. The Hubble parameter during the radiation dominated regime then reads

$H(T) = \sqrt{\frac{4 \pi^3 G}{45} g_* } \, T^2 = \sqrt{\frac{4 \pi^3}{45} g_* } \, \frac{T^2}{M_{Pl}}$

Remind that in RD epoch ρ ≈ a-4 and a(t) ≈ t1/2

### Thermal History of Early Universe at the time of RD

During RD epoch, which represents the earlier stage after inflation, several relevant phenomena occured. Some of them can be directly observed, wheras others are still in the domain of theoretical speculations, though they are quite strongly motivated on the basis of our present understanding of fundamental interactions. The main phenomena are the following:

1) Baryon asymmetry generation (Baryogenesis). The present value of the baryon/antibaryon asymmetry (no significant antibaryon density is measured in the observable universe if compared with the baryon one) requires that some baryon violating interactions took place in the early Universe. The temperature range for such a mechanism is likely to exceed the electroweak phase transition temperature, T ≈ 100 GeV.

2) Electroweak phase transition. At T ≈ 100 GeV the gauge symmetry SU(2)L x U(1)Y is spontaneously broken down to U(1)Q. At this scale all fermions, W± and Z0 become massive. Moreover both lepton and baryon numbers are also violated above this temperature, by non-perturbative effects.

### Thermal History of Early Universe at the time of RD

3) Quark-gluon  transition. At high temperatures quarks and gluons are not bound into hadrons. The hadronization takes place when the temperature drops below ΛQCD ≈ 200 MeV.

4) Neutrino decoupling and primordial nucleosynthesis. At T ≈  MeV, weak interactions  are no more efficient it keep neutrinos in equilibrium with e.m. plasma (corresponding interaction rates become smaller than the Hubble parameter), hence neutrinos decouple.  For the same reason chemical equilibrium between neutrons and protons is also lost. As a consequence neutron to proton density ratio reaches an almost constant asymptotic value, only affected by neutron decay. When T ≈ 0.08 MeV nuclear reactions start transforming relevant fractions of free nucleons into light nuclei.

### Matter Dominated (MD) Era

The RD epoch ends when non-relativistic matter becomes the main contribution to the total energy density. With the decreasing of the temperature due to the expansion, massive particles soon or later will become non-relativistic ( T ≈ m ). However, as long as equilibrium holds with other lighter species, the massive particle contribution is exponentially suppressed by the factor e-m/T. In this case these particles simply disappear from the plasma and release their entropy to lighter species. The only way for a massive particle to be the dominant contribution to energy is to escape from exponential suppression factor. Two ways to do it:

1) No more in chemical equilibrium with lighter species. In this case the collisional integral in the Boltzmann equation becomes negligible and the number density simply dilutes inversely proportionally to the comoving volume. The Weakly Interacting Massive Particle (WIMP) are an example of such a situation.

2) Massive particles are still in equilibrium, but carry a quantum number which is conserved by all interactions in the plasma. This means that their distribution has a chemical potential ξ associated with this charge such that

$n \sim (m T)^{3/2} e^{-m/T+\xi}$

The  chemical potential grows as temperature decreases and compensates the factor e-m/T.  This scenario applies to baryons (protons and neutrons at low energy). If we have an initial asymmetry baryons-antibaryons somehow produced in the early Universe and all interactions at later times preserve baryon, proton and neutron distributions develop a large (positive) chemical potential ξ at low temperature, compensating the exponential suppression factor.

### The equality point

If we define the parameter ηB as following

$\eta_B \equiv \frac{n_B-n_{\overline{B}}}{n_\gamma} \sim \frac{n_B}{n_\gamma}$

we see that after antibaryons disappearence it is just the ratio between baryon and photon number density and it remains constant as long as the photon number density scales as a-3. A value of η≈ 6 10-10 is in good agreement with both CMB and BBN. By using the present nγ and ρcr,0 we have

$\Omega_B = \frac{m_N n_B}{\rho_{cr,0}} = \eta_B \frac{m_N n_{\gamma,0}}{\rho_{cr,0}} \sim 0.365 \cdot 10^8 \eta_B\, h^{-2} \sim 0.02\, h^{-2}$

### The equality point

A relevant parameter is the equality point,  namely the value of the scale factor (aeq) ,  or redshift  (zeq) for which ρR= ρM. This is the onset of so-called Matter Dominated era and it is indirectly observable through the CMB anysotropies. From the present value of the photon temperature T0 = 2.725 K and of ρcr,0, we get Ωγ h2= 2.47 10-5 .Since the radiation energy density scales as (1+z)4, while the matter density evolves as (1+z)3, the equality pointis given by

$\Omega_\gamma h^2 (1+z_{eq}) = \Omega_M h^2 \, \,\,\, \Rightarrow \,\, 1+z_{eq} \sim 4 \cdot 10^4 \,\Omega_M h^2$

Baryons represent a subdominant contribution to the whole matter (non relativistic d.o.f.) energy density ΩM. The main contribution comes from Cold dark Matter (as will be discussed in the following), Ωdm. Hence from observations ΩM = Ωdm + ΩB = 0.13 h-2 .

### Relevant phenomena in MD epoch

The previous expression for zeq just gives an approximate estimate of the equality point since we have neglected in ΩR the contribution coming from relic neutrinos (Cosmic Neutrino Background or CNB). As will be discussed in the next it is easy to show that for temperature below the electron mass the whole radiation energy density reads

$\Omega_R h^2 = \Omega_\gamma h^2 \left( 1+ \frac{7}{8} 3 \left( \frac{4}{11} \right)^{4/3} \right)$

Neutrinos give a contribution of the same order of the photon one, hence we get

$1+z_{eq} = \frac{\Omega_M h^2}{\Omega_R h^2} = 2.4 \cdot 10^4 \Omega_M h^2 \sim 0.3 \cdot 10^4$

As long as it is possible to neglect the contributions coming from spatial curvature and from a cosmological constant we have

$H(z) = H_0 \sqrt{\Omega_M} (1+z)^{3/2} \left( 1 + \frac{1+z}{1+z_{eq}} \right)^{1/2}$

In MD epoch two important phenomena occurred :

1) When T ≈ 0.1 eV, i.e. around a redshift z ≈ 103, almost all electrons and protons recombine to form neutral hydrogen, so we have the so-called recombination. This is the time at which the image of the CMB that we see today is emitted.

2) When the Universe is matter dominated, structures on large scales start forming.

### Dark Energy (Cosmological constant) dominated era

In the late nineties different observation led to an unexpected result: the expansion of the Universe today and at small redshift, z ≤ 1 is accelerated. This evidence comes from the measurements of the luminosity distance dL performed by two groups on SNIa places at z ≈ 1. At these redshifts dL is sensitive to the deceleration parameter q.

Type Ia Supernovae allow to perform such measurement since they are consicdered a good example of standard candle. These objects result from explosive events taking place in binary systems where one of the two stars  is a white dwarf. Since the degenerate star accretes matter from its companion it becomes unstable when it reaches the limiting Chandrasekar mass. In this scenario, being the mass scale of such supernovae fixed by the Chandrasekar bound, their luminosity can vary only within a small range.

### Dark Energy (Cosmological constant) dominated era

A negative q (i.e. positive acceleration expansion ) is a sign of the presence of a dominant component that has P ≤ −ρ/3. In particular, a cosmological constant with P = − ρ represents a very good fit of the data.

According al these observations the current standard model of cosmology, called ΛCDM, assumes that the Universe has zero spatial curvature (k=0), a non-zero cosmological constant Λ, and a relevant presence of Cold Dark Matter (CDM). Of course, all this is in addition to the other well-known species (photons, leptons, baryons, neutrinos).

The results of the Supernova Cosmological Project, from (Knop et al., 2003). The effective apparent magnitude for 42 high redshift Type Ia Supernovae is plotted versus z and compared with different cosmological models

### The number of e-folds

To have an idea of the amount of inflation necessary to overcome horizon and flatness problems, let us remind that a comoving wavelength equal to the Hubble scale at the beginning of inflation should not re-enter inside this scale earlier than today. For this reason the minimum duration of inflation is given by the condition that the comoving wavenumber equal to

k= ain $\overline{H}$  is also equal to a0 H0

The Hubble parameter at ain and aend are almost the same  for a quasi-de Sitter expansion, moreover such value can be related to  H0 using the fact that when inflation ends, the Universe enters radiation domination, and switches to matter domination at the equality point aeq. Since H ≈ a-2 during radiation domination and H ≈ a-3/2 during matter domination,  we have

$\overline{H} \, \frac{a_{end}^2}{a_{eq}^2} \, \frac{a_{eq}^{3/2}}{a_{0}^{3/2}} =H_0$

Neglecting the impact of recent cosmological constant dominated epoch we have

$\frac{a_{end}}{a_{in}} \sim \frac{a_0}{a_{end}} \left(\frac{a_{eq}}{a_0} \right)^{1/2}$

### The number of e-folds

The number of e-folds, typically indicated with N, between two cosmological times is defined as the logarithm of the amount of expansion between these times. According to this notation N ≡ log(aend/ain) is the number of e-folds during inflation.

The condition obtained in the previous trasparency states that the minimum number of e-folds during inflation is given by the number of e-folds between the end of inflation and today. We can estimate the minimum value of N by specifying the value T of the temperature of relativistic species at the beginning of radiation domination,  since in very rough approximation T ≈ (a0/aend) T0.

This value cannot be smaller than the primordial nucleosynthesis scale T ≈ MeV, since an inflationary stage at this epoch would result in a much faster expansion rate. Nuclear processes would then be very inefficient, preventing helium production. This condition gives N > 19. More sophisticated arguments can be invoked to straighten this bound, but they usually depend on which extension of the standard model of particle physics is assumed.

Note that during inflation, the contribution of curvature to the expansion, given by Ωk=1-Ω, is reduced by a factor e-2N.

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